Problem 12 - Algebra 2

Question

Write each expression in radical form. $$ x^{\frac{2}{7}} $$

Step-by-Step Solution

Answer

The exponential expression \(x^{2/7}\) can be written in radical form as \(\sqrt[7]{x^2}\).

Step 1: Identify the base, numerator, and denominator

The base of the exponent is \(x\), the numerator of the fractional exponent is \(2\), and the denominator is \(7\). The base will remain the base in the radical expression, and the numerator and denominator of the fractional exponent would correspond to the power of the radicand and the index of the root respectively.

Step 2: Write the expression in radical form

With the above understanding, the expression \(x^{2/7}\) can be rewritten in radical form as \(\sqrt[7]{x^2}\).

Key Concepts

Fractional Exponents

Fractional exponents, often seen in expressions like \(x^{\frac{2}{7}}\), are an interesting way to represent roots and powers simultaneously. These exponents have two parts, a numerator and a denominator, usually written as a fraction.

• The numerator denotes the power to which the base is raised.
• The denominator indicates the root to be taken.

For instance, consider the expression \(x^{\frac{p}{q}}\). Here, \(p\) is the power and \(q\) signifies the root. When rewritten in radical form, it becomes \(\sqrt[q]{x^p}\), expressing the same operation in a different format. This method is versatile for handling complex algebraic operations more conveniently.

Radicand

The term radicand refers to the quantity inside a radical symbol, such as the expression \(\sqrt[7]{x^2}\). In our exercise, the base \(x\) becomes the radicand when translating the fractional exponent into radical form.

• The radicand is what is subject to the root operation.
• In \(\sqrt[7]{x^2}\), \(x^2\) is the radicand part.

The radicand does not change the essential property of the expression. However, it specifies exactly what is being manipulated by the radical operation, crucial for solving and simplifying mathematical expressions involving roots.

Index of a Root

In expressions using radical symbols, the index of a root describes which root is being taken. For \(x^{\frac{2}{7}}\), the denominator \(7\) in the fractional exponent indicates the 7th root, making the expression \(\sqrt[7]{x^2}\).

• The index is typically denoted by a small number written near the upper left of the radical symbol.
• If no index is shown, it is understood to be a square root, or 2nd root.

Understanding the index is vital in expressions requiring accurate mathematical operations, as it defines the level of recurrence necessary to compute the root.

Base in Exponents

The base in exponents, particularly in expressions explaining fractional exponents, is the fundamental quantity that is raised to a power or is the subject of the root operation. In the exercise \(x^{\frac{2}{7}}\), \(x\) serves as the base.

• The base is the central figure of operation whether the context involves multiplication (as in powers) or division (as in roots).
• No matter the transformation into radical form or the complexity, the base remains integral to the expression.

When dealing with powers and roots, identifying the base is a primary step. It helps in tracking the primary component of an expression undergoing complex algebraic manipulations.